The computation of dimensions of linear systems of singular hypersurfaces
is a difficult problem that has recently attracted much attention from
algebraic geometers, mainly when the singularities are in general
position. One of the most successful approaches to this problem, developed
by several authors, consists in considering families of such linear
systems having a special member with fixed components that can be split
off. This allows to reduce to another system of hypersurfaces of lower
degree and simpler singularities, and in the best cases to compute
inductively the desired dimensions.
In the talk we discuss this approach and the results that can be derived
from it, focusing on applications to the Harbourne-Hirschowitz conjecture
that we have obtained recently.